Theorem nfrex2w-P6 695

Description: ENF Over Existential Quantifier (different variable - weakened form).

Requires the existence of '𝜑₁(𝑥₁)' as a replacement for '𝜑(𝑥)'.

Hypotheses

Ref	Expression
nfrex2w-P6.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
nfrex2w-P6.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrex2w-P6	⊢ Ⅎ𝑥∃𝑦𝜑

Distinct variable groups:   𝜑,𝑥₁   𝜑₁,𝑥   𝑥,𝑦,𝑥₁

Proof of Theorem nfrex2w-P6

Step	Hyp	Ref	Expression
1		nfrex2w-P6.1	. . . . 5 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	subexv-P5 624	. . . 4 ⊢ (𝑥 = 𝑥₁ → (∃𝑦𝜑 ↔ ∃𝑦𝜑₁))
3	2	subneg-P3.39 323	. . 3 ⊢ (𝑥 = 𝑥₁ → (¬ ∃𝑦𝜑 ↔ ¬ ∃𝑦𝜑₁))
4	1	lemma-L5.05a 668	. . . . 5 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
5	1	subneg-P3.39 323	. . . . . . . 8 ⊢ (𝑥 = 𝑥₁ → (¬ 𝜑 ↔ ¬ 𝜑₁))
6		nfrex2w-P6.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜑
7		nfrneg-P6 688	. . . . . . . . 9 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
8	6, 7	bimpr-P4.RC 534	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝜑
9	5, 8	nfrgenw-P6 684	. . . . . . 7 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
10		lemma-L5.01a 600	. . . . . . 7 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
11	9, 10	bimpr-P4.RC 534	. . . . . 6 ⊢ (∃𝑥𝜑 → 𝜑)
12	11	alloverimex-P5.RC.GEN 603	. . . . 5 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑦𝜑)
13	4, 12	syl-P3.24.RC 260	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦𝜑)
14		lemma-L5.01a 600	. . . 4 ⊢ ((∃𝑥∃𝑦𝜑 → ∃𝑦𝜑) ↔ (¬ ∃𝑦𝜑 → ∀𝑥 ¬ ∃𝑦𝜑))
15	13, 14	bimpf-P4.RC 532	. . 3 ⊢ (¬ ∃𝑦𝜑 → ∀𝑥 ¬ ∃𝑦𝜑)
16	3, 15	gennfrw-P6 685	. 2 ⊢ Ⅎ𝑥 ¬ ∃𝑦𝜑
17		nfrneg-P6 688	. 2 ⊢ (Ⅎ𝑥 ¬ ∃𝑦𝜑 ↔ Ⅎ𝑥∃𝑦𝜑)
18	16, 17	bimpf-P4.RC 532	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by: (None)